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• Title Page
• Copyright
• Preface
• PART I
• chapter 1. LANGUAGE, LOGIC, AND SETS
• 1.1. Logic and Language
• 1.2. Implication
• 1.3. Quantifiers and Definitions
• 1.4. Introduction to Sets
• 1.5. Introduction to Number Theory
• 1.6. Additional Set Theory
• Definitions from Chapter 1
• Algebraic and Order Properties of Number Systems
• chapter 2. PROOFS
• 2.1. Proof Format I: Direct Proofs
• 2.2. Proof Format II: Contrapositive and Contradiction
• 2.3. Proof Format III: Existence, Uniqueness, Or
• 2.4. Proof Format IV: Mathematical Induction
• The Fundamental Theorem of Arithmetic
• 2.5. Further Advice and Practice in Proving
• Proof Formats
• chapter 3. FUNCTIONS
• 3.1. Definitions, Notation, and Examples
• 3.2. Composition, One-to-One, Onto, and Inverses
• 3.3. Images and Pre-Images of Sets
• Definitions from Chapter 3
• chapter 4. RELATIONS
• 4.1. Relations
• 4.2. Equivalence Relations
• 4.3. Partitions and Equivalence Relations
• 4.4. Partial Orders
• Definitions from Chapter 4
• PART II
• chapter 5. INFINITE SETS
• 5.1. The Sizes of Sets
• 5.2. Countable Sets
• 5.3. Uncountable Sets
• 5.4. The Axiom of Choice and Its Equivalents
• Definitions from Chapter 5
• chapter 6. INTRODUCTION TO DISCRETE MATHEMATICS
• 6.1. Graph Theory
• 6.2. Trees and Algorithms
• 6.3. Counting Principles I
• 6.4. Counting Principles II
• Definitions from Chapter 6
• chapter 7. INTRODUCTION TO ABSTRACT ALGEBRA
• 7.1. Operations and Properties
• 7.2. Groups
• Groups in Geometry
• 7.3. Rings and Fields
• 7.4. Lattices
• 7.5. Homomorphisms
• Definitions from Chapter 7
• chapter 8. INTRODUCTION TO ANALYSIS
• 8.1. Real Numbers, Approximations, and Exact Values
• Zeno’s Paradoxes
• 8.2. Limits of Functions
• 8.3. Continuous Functions and Counterexamples
• Counterexamples in Rational Analysis
• 8.4. Sequences and Series
• 8.5. Discrete Dynamical Systems
• The Intermediate Value Theorem
• Definitions for Chapter 8
• chapter 9. METAMATHEMATICS AND THE PHILOSOPHY OF MATHEMATICS
• 9.1. Metamathematics
• 9.2. The Philosophy of Mathematics
• Definitions for Chapter 9
• THE GREEK ALPHABET
• ANSWERS SELECTED ANSWERS
• INDEX
• LIST OF SYMBOLS